Torsion

ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 98-S44

Torsion of High-Strength Reinforced Concrete Beams and Minimum Reinforcement Requirement by Nasr-Eddine Koutchoukali and Abdeldjelil Belarbi To study the effect of high-strength concrete on the torsional behavior of reinforced concrete (RC) beams, nine full-size beams were tested under pure torsion. The main parameters in this study were concrete strength and amount of reinforcement. Concrete strength ranged from normal strength through all grades of highstrength concrete (defined as 50, 60, 80, and 95 MPa). The amount of reinforcement varied from less than the minimum to the socalled balanced condition (when expected crushing of concrete occurs at the same time as yield of steel). With the intent of keeping the inclination of the concrete struts approximately equal to 45 degrees, equal percentages of reinforcement were provided in the transverse and longitudinal directions. Results indicate that the minimum amount of reinforcement defined in ACI 318-99 is inadequate for equilibrium torsion of high-strength RC beams, and a new expression is proposed. It was found that the torsional capacity of under-reinforced beams is independent of concrete strength, and the amount of longitudinal reinforcement was more effective in controlling crack width than the amount of transverse reinforcement (stirrups).

Fig. 1—Examples of structures where torsion is predominant action.

Keywords: Keywords : beam; concrete, high-strength; reinforcement; shear; strain; torsional stress.

INTRODUCTION Torsion can become a predominant action in structures such as eccentrically loaded box beams, curved girders, spandrel beams, and spiral staircases (Fig. 1). Prior to 1995, the design and analysis of such members were based on semi-empirical semi-empirical provisions and were lacking rationality. Furthermore, prestress was not addressed in torsion design. In 1995, ACI 318-951 adopted new torsion provisions that seem to be more rational. This new method is based on the thin-wall tube/space truss analogy and is capable of addressing both reinforced and prestressed concrete. In this method, the torsional member is idealized as a tube that, after cracking, becomes a space truss where transverse (stirrups) and longitudinal reinforcement reinforcement are in tension and concrete diagonals are in compression (Fig. 2). Tests and theoretical interpretation 2,3 have shown that once cracking has occurred, the concrete in the center of the member contributes little to the torsional strength of the cross section and can thus be ignored. In previous versions of ACI building code (from 1971 to 1989), torsional strength of beams was considered to be composed of two parts: the concrete contribution T c, and the reinforcement contribution T s. While the torsional moment strength provided by reinforcement T s was obtained from the equilibrium of a space truss, assuming the diagonal compression struts to have an inclination of 45 degrees, the procedure included a correction factor αt (empirical) that incorporates the effect of the cross-section aspect ratio. In ACI 318-95, torsional moment strength provided by concrete was eliminated. Furthermore, the simplification in the estimate of the

462

Fig. 2—Space truss analogy.

lever arm area Ao (Fig. 2) did not address the cross-section aspect ratio. Torsional design provisions are based on yield of steel, meaning that design is based on the under-reinforced beam concept. All test data used to validate this method were based on beams having a maximum concrete strength of approximately 50 MPa. 4 Very few tests of high-strength reinforced concrete (RC) beams under torsion have been reported in the literature. 5 For this reason, ACI 318-95 limits the use of the theory up to a concrete strength f c′ = 69 MPa. This paper reports the results of an investigation on torsional behavior of RC beams6,7 with emphasis on high-strength concrete as a contribution to the understanding of the behavior of such members and their failure modes, and it addresses some design issues specific to pure torsion of high-strength RC members, such as minimum reinforcement and effects of concrete strength. ACI Structural Journal, V. 98, No. 4, July-August 2001. MS No. 00-132 received June 2, 2000, and reviewed under Institute publication policies. Copyright © 2001, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the May-June 2002 ACI Structural Journal if received by January 1, 2002.

ACI Structural Journal/July-August 2001

ACI member Nasr-Eddine Koutchoukali is a project engineer with HBE Corp., St. Louis, Mo., Mo., and a former graduate graduate research research assistant assistant in civil civil engineering engineering at the the University of Missouri-Rolla. He received his BSE from the University of Constantine, Algeria, his MS from the University of Washington, Seattle, and his PhD from the University of Missouri-Rolla. Missouri-Rolla. His His research research interests interests include include nonlinea nonlinearr behavior behavior of reinforced reinforced concrete concrete members, seismic retrofit of structures, and structural renovation of buildings. Abdeldjelil Belarbi , FACI, is an associate professor of civil engineering, University of Missouri-Rolla, Mo.

RESEARCH SIGNIFICANCE Studies with experimental evidence are very limited for torsion of high-strength concrete beams. The present experimental investigation addresses the existing research void pertaining to the torsional behavior of high-strength, underreinforced RC beams. With the increased use of highstrength concrete in structures such as bridges, where torsion can be an important design factor, it seems necessary for any design code to include commercially produced concrete strengths up to 100 MPa. Design equations for minimum reinforcement for torsion in the present code are discussed in this paper, and changes are proposed for equilibrium torsion.

EXPERIMENTAL PROGRAM Specimen details In torsion, beams can be: 1) under-reinforced, when both the stirrups and the longitudinal reinforcement yield before crushing of concrete diagonals; 2) partially over-reinforced, when either stirrups or longitudinal bars, but not both, yield before crushing of concrete diagonals; and 3) over-reinforced, when crushing of concrete takes place before steel yields. Furthermore, a minimum amount of reinforcement should be provided in both directions to ensure the post-cracking capacity and desirable postyielding ductility. Because the present torsional provisions are based on yield of steel, the RC beams included in this test series were designed to be underreinforced or near the balanced condition (when crushing of concrete occurs at the same time as yield of steel). All test beams were 3.96 m long, having a cross section of 305 by 203 mm (Fig. 3). A concrete cover of 19 mm was used. To allow for failure to occur in the central test region of the beam, additional stirrups were placed at both ends of the beam. The test region was selected to be 2.38 m long to allow at least two complete spirals of diagonal cracks to form along the beam and also to accommodate the test-rig length in the laboratory. This made each of the two heavily reinforced ends 0.8 m long. Two series of rectangular, full-size beams were tested. The investigated parameters were concrete strength for Series 1, and amount of reinforcement for Series 2. Series 1 included five beams with design strengths selected to be 35, 48, 62, 83, and 96 MPa. This range was chosen to cover most grades of normal- and high-strength concrete. 8 Identification of the individual test beams is given by the form BXURN, where X stands for the design concrete strength of a given beam in ksi, UR refers to under-reinforced, and N is the beam number of the same concrete strength, ordered from lowest to highest reinforcement ratio. Series 2 included four beams having a constant concrete strength (approximately (approximately 76 MPa) and a total reinforcement ratio that varied from 1.76 to 2.64%. Table 1 summarizes the characteristics of reinforcement and concrete for all beams included in this investigation.


Fig. 3—Cross-section and reinforcement arrangement of test beam of first series (dimensions in mm).

Material properties Table 2 gives the mixture proportions for the various concrete strengths. Local river sand had a fineness modulus of 2.3. Two different local crushed aggregates were used: a 12.7 mm maximum aggregate size limestone for the three f c′ equal to 35, 48, and 62 MPa; and a 12.7 mm mixtures of f maximum aggregate size granite for the two mixtures of f c′ equal to 83 and 96 MPa. A melamine-based high-range wat erreducing admixture (HRWR) was used for the 62 MPa concrete, and a modified naphthalene-sulfonate-based HRWR was used for the 83 and 96 MPa mixtures. Type I portland cement was used for all five mixtures, Type C fly ash was used in the 48 MPa concrete, and silica fume in powder form was used for the 83 and 96 MPa mixtures. An air-entraining agent (AEA) was used in the 35 MPa mixture to reduce bleeding water. Typically, six to 12 100 x 200 mm cylinders and six to 12 152 x 305 mm cylinders were cast with each test beam and used for compression and splitting tests. Beams B5UR1, B7UR1, and B9UR1 were made of two similar batches. The rest of the beams were made from three batches to allow for smaller mixture quantities and better handling. A few hours after casting, the beams and cylinders were covered with wet burlap and plastic sheets. Form stripping was done about 24 h after casting, and both beams and cylinders were kept under the same curing conditions. Compression and splitting tests on cylinders were carried out around the time of beam testing. Table 1 gives the test results of the cylinders as well as the age at testing (some specimens were tested beyond the aimed 28 days because of scheduling issues). Strengths of the 100 x 200 mm cylinders were taken as the reference strength for all beams except B5UR1, which was based on 152 x 305 mm cylinder results.

Test setup and instrumentation instrumentation Details of the test setup are shown in Fig. 4. One 267 kN hydraulic actuator was used to apply the load near the east support. The load had a 457 mm lever arm from the centroidal axis of the beam, giving the test rig a 122 kN-m torque capacity. A 445 kN tension load cell was used to measure the applied load. The actuator had a stroke length of 178 mm providing a 19 degree twist capacity of the beam. A reaction arm was used near the west support to balance the applied load by attaching the arm to the laboratory strong floor. The reaction rod had a 457 mm eccentricity from the centroidal axis of the beam as well. After cracking, the beam elongates

463

Table 1—Material properties Longitudinal reinforcement

d v , mm

f yv, MPa

s, mm

B5UR1

4 No. 4

386

9 .5

373

108

30

39.6

3 .6

B7UR1

4 No. 4

386

9 .5

399

108

34

64.6

4 .5

Testing age, days f c′, MPa

f ct , MPa

B9UR1

4 No. 4

386

9 .5

373

108

35

75.0

4 .6

B12UR1

4 No. 4

386

9. 5

399

108

54

80.6

5 .3

B14UR1

4 No. 4

386

9. 5

386

108

56

93.9

6 .3

386

102

51

76.2

5 .5

386

95

90

72.9

5 .3

B12UR2 2

Concrete

f yl , MPa

Series Series no. Test Test beam beam

1

Transverse reinforcement

No. of bars

4 No. 4

386

9. 5

4 No. 4

373

9. 5

2 No. 3

386

9. 5

B12UR4

6 No. 4

373

9. 5

386

90

48

75.9

5. 3

B12UR5

4 No. 5

380

9. 5

386

70

81

76.7

5. 5

B12UR3

Table 2—Concrete mixture proportions Concrete mixture proportions Constituents

35 MPa 48 MPa 62 MPa 83 MPa 96 MPa

Cement, kg/m3

345

508

613

568

503

Water, kg/m3

193

191

193

138

138

0.56

0. 3 8

0.31

0.24

0. 2 7

847

634

586

693

649

831

1036

976

1112

1112

Fly ash, kg/m3

—

61

—

—

—

Silica fume, kg/m3

—

—

—

41.5

47.5

w / c 3

Fine aggregate, kg/m

Coarse aggregate, kg/m

Fig. 4—Test setup for pure torsion.

Fig. 5—Location of electrical strain gages (dimensions in mm).

longitudinally. To avoid any longitudinal restraint and subsequent compression, the beam was allowed to slide and elongate freely. This was achieved by supporting the west end of the beam on rollers. In the beams of Series 1, 15 electrical resistance strain gages were used to measure strains on the reinforcing bars. Six strain gages were mounted on three stirrups within the test region, with one stirrup located at midspan and two symmetrically located at 648 mm from midspan. Each stirrup was instrumented with two strain gages, one mounted at the middle of the short leg (bottom face) and one at the middle of the long leg (side face; Fig. 5). Nine strain gages were mounted on longitudinal bars at three different sections of

464

3

Water reducer, mL

—

745

—

—

—

High-range water reducing admixture, L

—

—

3.752

2 4. 3

21.0

Air-entraining admixture, mL

130

—

—

—

—

Slump, cm

1 1. 5

11.5

7. 5

Flowing Flowing

the test region. One set of three gages was located in the middle, and the other two sets symmetrically symmetrically located at 648 mm from the middle on each side of the test beam. At each section, two gages were mounted on the bottom corner bars and one gage on an upper corner bar. In beams of Series 2, all details of the instrumentation of the test beams were the same as those of the beams from Series 1, with one exception: the number of strain gages used to measure strains in the longitudinal bars was reduced by one (eight strain gages instead of nine). Additionally, to measure the thickness of the shear flow zone, multilayered strain gage units developed at University of Missouri-Rolla 9 were embedded in the concrete and used to measure compressive strains through the thickness of the concrete strut at the outer skin of the beam and toward the inside of the beam. Two units of three multilayered strain gages were placed at midspan of the beam: one mounted at the middle of the long side, and the other at the middle of the short side of the beam’s cross section. When additional longitudinal bars were used at midheight of the long side (for Beams B12UR3 and B12UR4), these units—at midheight of the long side of the beam—were shifted slightly upward. Figure 6 shows one of these units and describes the location on test beams. The twist of the beam was measured by a rotational variational differential transducer (RVDT) having a gage length of 1.93 m and located in the middle of the test region. Linear


variable differential transformers (LVDTs) were used to measure concrete surface strains as well as beam end elongation. Three LVDTs were placed in a rosette format to measure average concrete strains in three directions as shown in Fig. 7. One LVDT was placed along the longitudinal axis of the beam and the other two, at 45 and 135 degrees, measured counter-clockwise from the longitudinal axis. A fourth

LVDT was placed near the support to measure the longitudinal elongation of the beam (Fig. 7).

Test procedure Measuring devices for load, deformation, and strains were read through a computer-driven data acquisition system. Prior to failure of the beam, data were recorded at a prescribed load increment. Smaller increments increments were used around cracking crackin g to accurately measure the value closest to actual cracking torque. At every load stage after cracking, the load was held constant for several minutes before collecting the data, after which the crack pattern was marked, crack width and spacing were measured, and concrete spalling-off was checked. This check was performed by knocking on the concrete with a steel hammer and listening to the sound. A hollow sound would indicate separation of the concrete cover from the concrete inside the stirrups. When the beam reached its torsional capacity, data was continuously recorded until the hydraulic jack reached its maximum stroke, which corresponds to the maximum twist capacity of the setup.

TEST RESULTS AND DISCUSSIONS Cracking characteristics Fig. 6—Embedded strain gages and their location on test beams (dimensions in mm).

An accurate estimate of cracking torque is important in the case of compatibility torsion. In indeterminate structures, torsional moment can be redistributed to the adjoining members after cracking. 10, 11 For this reason, the ACI Code allows

Fig. 7—Different deformation measurements of test beam.

Table 3—Experimental results At cracking

At ultimate –2

Test beam

Torque, kN-m

Twist, 10 degrees/m

Torque, kN-m

Twist, degrees/m

B5UR1

11.6

8 .5

1 9. 4

2. 1 8

B7UR1

14.1

8. 0

1 8. 9

1. 9 3

B9UR1

13.0

13.4

2 1. 1

2. 0 5

−3000

Not measured

1450

B12UR1

16.2

9 .1

1 9. 4

1. 1 4

1248

19.3

11.7

2 1. 0

0. 2 1

260

233

B12UR2

17.8

1 1. 1

1 8. 4

0. 7 5

5355

377

B12UR3

16.0

1 0. 2

2 2. 5

2. 0 2

12,281

1658

B12UR4

16.9

14.0

2 3. 7

1. 9 3

7065

1425

B12UR5

13.6

3.56

2 4. 0

2. 5 0

−1940 −292 −1488 −2578 −1971 −2650

3218

B14UR1

9772

1733


εds, 10–6 m/m εr , 10–6 m/m εl , 10–6 m/m 5010 1505 −1819 5827 1488 −2441

46 5

Table 4—Comparison of experimental results with ACI predictions At cracking Beam

T cr,ACI , kN-m T cr,test / T T cr,ACI T n,ACI , kN-m T n,test / T T n,ACI

B5UR1

7. 9

1. 4 7

16.4

1.17

B7UR1

10.1

1. 4 1

1 7. 1

1.10

B9UR1

10.9

1. 2 0

1 6. 5

1.28

B12UR1

11.3

1. 4 4

17.1

1.14

B14UR1

12.1

1. 5 9

17.1

1.23

B12UR2

10.9

1. 6 3

17.1

1.06

B12UR3

10.7

1. 5 0

20.0

1.11

B12UR4

10.9

1. 5 5

23.8

1.00*

B12UR5

11.0

1. 2 4

24.0

1.00*

1. 4 5

—

1.12

Mean *

At ultimate

Based on crushing of concrete.

the torsional moment to be reduced to the cracking torsional moment in members where redistribution of internal forces is possible. Under combined stresses, ACI 318-99 provides an estimate of cracking torsional moment as

(a)

2

T cr

A = 0.33 f c′ -----c-- p c

(1)

in which the units are MPa and mm. Up to cracking, the behavior of the beam is essentially elastic. Torsion is mainly resisted by concrete. Prior to cracking, the measured surface concrete strains along the diagonal at 45 and 135 degrees (Fig. 7) are essentially of the same magnitude, but of the opposite sign, with tension being along the 135 degree direction. Measured values of torque and twist at cracking are shown in Table 3. Table 4 provides the calculated cracking torque of the nine tested beams based on Eq. (1). The mean value of the ratio of the measured torque at cracking to the calculated torque is 1.45 with a coefficient of variation of 0.096. This leads to the conclusion that the ACI Code underestimates the cracking torque for pure torsion by about 31%. Similar conclusions were reported by Ghoneim and MacGregor, 4 based on 94 torsional members. They found that the Code underestimates the cracking torque by approximately 32%.

(b) Fig. 8—Torque-twist relationships for tested beams: (a) Series 1; and (b) Series 2.

Postcracking behavior After cracking, concrete behaves as a nonlinear and discontinuous medium that leads to redistribution of internal stresses, forming a truss action in which reinforcement acts as tensile links, and concrete acts as compression diagonals. As applied torque increases, spiral cracks develop at about 45 degrees and spread over the test region. Because the beams were reinforced with equal amounts of reinforcement in the longitudinal and transverse directions, all cracks were inclined at 45 degrees throughout the loading history. These cracks were evenly distributed in the test region, except for Beam B14UR1, which experienced one major crack (followed by failure), and Beam B12UR2, which experienced several cracks in the central region of the beam but failed before cracks became evenly distributed along the beam. Such behavior in the latter beams is attributed to the fact that these beams were reinforced with an amount of reinforcement close to the minimum, and limited strength after cracking

466

was available. This reduced the amount of cracking along the beam as shown in Table 3 by the smaller amount of strain recorded in the longitudinal direction εl Ultimate torque and twist —Torque, —Torque, twist, and concrete surface strains are given in Table 3 for all test beams. For the first series, results show that as concrete strength increases, behavior changes from barely under-reinforced (close to balanced) to grossly under-reinforced (close to minimum reinforcement). For Beam B5UR1, yield of steel occurred just a few load stages before failure of the beam (crushing of concrete), whereas for B14UR1, yield of steel occurred just after cracking of the beam. Although these beams were similarly reinforced, the torque at cracking for Beam B14UR1 was high, and the amount of reinforcement provided did not supply sufficient strength after cracking to prevent a brittle failure. The data recorded indicate that the ratios of ultimate to cracking .


Fig. 10—Concrete internal strain showing linearly varying strain distribution and thickness of shear flow zone.

Fig. 9—Torque-maximum crack width relationships of test beams.

torque for Beams B12UR1 and B14UR1 were only 1.2 and 1.09, respectively. The torque-twist relationships for beams of both series are shown in Fig. 8. For the second series, the behavior of beams ranges from minimum reinforcement (Beam B12UR2 with limited ductility) to slightly over-reinforced (Beams B12UR4 and B12UR5 with failure by crushing of concrete). For both Beams B12UR4 and B12UR5, yield of steel was impending at failure, meaning that the amount of reinforcement provided was close to the balanced condition. Equation (2) provides the ACI 318-95 estimate of torsional strength of under-reinforced RC beams. Table 4 gives a comparison between the experimental ultimate torques T n,ACI and those predicted by Eq. (2). The mean T max,test / T for all beams is 1.12, which shows that the ACI Code underestimates the ultimate strength by approximately 11%. These results indicate that the ACI 318-95 estimate for torsional strength is reasonably conservative for these tested beams. 2 A o At f ------- yv --- cot θ T n, A C I = ------------s

(2)

where cotθ is defined as

cot θ =

Al f y l s ---------------- At f yv p h

(3)

Crack width and failure modes of beams—Test results from the first series showed that when the amount of reinforcement is kept constant, the initial crack width increases with the increase of concrete strength (Fig. 9), which induces larger strains in the reinforcing steel at beam cracking. As torque approaches the ultimate capacity of the beam, and shortly after yield of the longitudinal reinforcement, the width of the main crack (usually the first one to form) increases rapidly and causes the longitudinal bar to kink, resulting in a sudden drop in the capacity of the beam. At this stage of the test, all de-


formations take place at the location of the main crack, where the rest of the beam is in an unloaded state. This phenomenon is probably more severe for high-strength RC beams than for normal-strength RC beams because the smooth-faced cracks are less effective in transmitting shear stresses for the former. Concrete internal strains—As expected, the measured concrete strains through the cross section showed two distinct stages, before and after cracking. Before cracking, all measured strains in the concrete strut at 45 degrees counterclockwise from the longitudinal direction were compressive strains. After cracking, the innermost strain gage measured tension, suggesting warping of the concrete strut. As assumed by the theory, measured strains varied linearly through the thickness of the shear flow zone (Fig. 10).

Minimum torsional reinforcement for equilibrium torsion A minimum amount of reinforcement is needed to ensure that the beam does not fail at cracking. Observed behaviors of Beams B12UR1 and B14UR1 provide valuable information regarding minimum reinforcement for high-strength concrete beams. Although Beam B14UR1 was reinforced with an amount much higher than that prescribed by the ACI Code, the beam failed at a very small twist. This shows a brittle failure after cracking, meaning that there was not enough reinforcement for postcracking strength. Observation of Beam B12UR1 also shows the following: to allow for the truss action to form and to develop postcracking behavior as opposed to local brittle failure, the beam needed about 20% reserve strength after cracking. This is probably due to a larger crack width at cracking for high-strength RC beams than for normal-strength RC beams (Fig. 9) and to the previously mentioned smooth-faced-cracks that are less effective in transmitting shear stresses. Based on these observations, the following procedure is adopted to derive the minimum amount of reinforcement for torsion. From the data obtained in this study, corroborating the average obtained by Ghoneim and MacGregor, 4 an estimate of cracking torque is given by 2

T cr

A = 0.46 f c′ -----c-- p c

(4)

Assuming a 45-degree angle for θ, 20% reserve in capacity after cracking as in Beam B12UR1 (that is, T n / T Tcr = 1.2), and equating the right-hand sides of Eq. (2) and (4) times λ Tcr = 1.2, gives = T n / T

46 7

2

f c ′ A c  A-----t  = 0.28 ------------------ s  mi n f yv A o p c

(5)

is independent of concrete strength, 14 as assumed in the truss model adopted in ACI 318-95.

CONCLUSIONS AND RECOMMENDATIONS RECOMMENDATIONS For θ equal to 45 degrees A t f yv A l = ----- p h -----s f y l

(6)

and replacing At / s by the right-hand side of Eq. (5) results in 2

( Al )mi n

f c′ p h A c = 0.28 --------- ----- ------- f y l P c A o

(7)

Use of Eq. (5) and (7) removes the unnecessary confusion that can face a designer when using Eq. (11-24) of ACI 318-95, as shown by Ali and White. 12 The amount of minimum reinforcement proposed in this study is similar to that proposed by Ali and White. Though the ratio λ was taken as 1.2 in this study (as compared with 1.7 in Ali and White), this was compensated by the assumed magnitude of torsional torque. The authors assumed a magnitude of T cr proposed by Ghoneim and MacGregor, 4 while Ali and White used the magnitude provided by the ACI Code. The former is about 40% larger than the latter. It should be noted that the ACI cracking torque estimate is based on beams subjected to combined stresses, whereas the present results are for beams subjected to pure torsion. For lightly reinforced beams, comparison of the behaviors of Beams B12UR2 and B12UR3 shows that longitudinal reinforcement is more effective than transverse reinforcement in increasing the torsional capacity of the beam. This may be attributed to the contribution of the longitudinal bars, which, in keeping crack width smaller, is probably more significant for high-strength concrete than for normal-strength concrete because of the reduced contribution of aggregate interlock to shear. Furthermore, the theory is based on the smeared reinforcement concept as follows: for small beams like those tested in this study, it is inappropriate to assume that longitudinal reinforcement can be visualized as smeared when only corner bars are used for reinforcement. Therefore, transverse reinforcement should be designed to yield prior to longitudinal reinforcement to avoid large crack widths. Achieving a larger amount of longitudinal than transverse reinforcement in design can be accomplished by requiring an angle θ smaller than 45 degrees.

Effect of concrete strength Prior to 1995, in the ACI Code, a contribution by the concrete compression zone acting in torsional shear T c was included in the torsional capacity of RC beams. Test results from PCA and the University of Stuttgart investigations 2,13 support this inclusion. The strength of the concrete used in those tests ranged approximately from 15 to 45 MPa and 27 to 53 MPa, respectively. Based on the results of PCA investigation, early versions (from 1971 until 1989) of the Code included a concrete contribution T c that was taken to be proportional to √ f c′. The results from the first series of this investigation, however, in which concrete strength ranged approximately from 40 to 94 MPa, do not substantiate such findings and tend to support the notion that torsional strength

468

Results of this study lead to the following conclusions: 1. The minimum amount of reinforcement provided in ACI 318-95 is inadequate for high-strength RC beams. To avoid brittle failure (at the onset of cracking), a 20% reserve of strength after cracking must be available for the beam to experience uniform cracks along its length. Based on this assumption, a new expression for minimum torsional reinforcement was developed; 2. Torsional strength of RC beams is independent of concrete strength as long as the beam is under-reinforced (steel yielding); and 3. The observed failure of high-strength RC beams was mainly controlled by the amount of strain of longitudinal reinforcement. When larger crack widths develop at yield of the longitudinal reinforcement, the beam forms a torsional hinge at the yield/crack location and ceases on behaving as a unit. Localized twist increases at the hinge region, whereas other regions of the beam undergo an unloading process. This is probably due to the smooth-faced cracks, which are less effective in transmitting shear stresses and contribute to early failure of beams. Further research effort remains to be implemented to accurately predict the behavior of high-strength RC beams. The effects of parameters such as size, reinforcement ratios, and combined actions (torsion, shear, and bending moments) still need to be clarified.

CONVERSION FACTORS 1 kg/m3 1 kN 1 k N- m 1 mm 1 MP a

= = = = =

1.6 86 86 lb /y /yd3 0.225 kip 8.849 kip-in. 0.0394 in. 145 psi

NOTATION Ac A l Ao Aoh

= = = =

At f c′ f yl f yv pc ph s

= = = = = = =

T c T cr T n T s

= = = =

t d = V 1 to V 4 = xo = yo

=

εds

=

εl εr

= =

θ λ

= =

area enclose enclosed d by outsi outside de perime perimeter ter of of concret concretee crosscross-sect section ion total total area area of of longit longitudin udinal al reinf reinforce orcement ment to resi resist st torsi torsion on gross gross area area encl enclos osed ed by shea shearr flow flow path path area enclose enclosed d by centerli centerline ne of of outermo outermost st close closed d transv transverse erse torsional reinforcement area of one leg of closed stirrup resisting torsion within spacing s conc concre rete te com compr pres essi sive ve str stren engt gth h yield yield stre streng ngth th of of longit longitudi udina nall reinf reinforc orceme ement nt yield yield streng strength th of of closed closed transver transverse se torsi torsional onal reinforc reinforcemen ementt perim perimete eterr of concre concrete te cross cross secti section on perimeter of centerline of outermost closed stirrup spacing spacing of transv transverse erse reinforc reinforcemen ementt in directio direction n parall parallel el to to longitudinal reinforcement nominal torsional moment strength provided by concrete crac cracki king ng tor torsi sion onal al mome moment nt of of beam beam nomi nomina nall tors torsio iona nall mome moment nt str stren engt gth h nominal nominal torsion torsional al momen momentt stren strength gth provide provided d by torsion torsion reinforcement wall wall thick thickne ness ss of of equiv equival alen entt thinthin-wal walled led tube tube shear shear force forcess in Sides Sides 1 to 4 of space space truss truss due due to torsion torsion center-to-center length of shorter side of closed rectangular stirrup centercenter-to-c to-cente enterr length length of longer longer side side of closed closed rectan rectangula gularr stirrup surface surface strain strain in diagonal diagonal concrete concrete struts struts (45 (45 degree degreess from from longitudinal axis of beam) surface surface concrete concrete strain strain along along the longitud longitudinal inal axis of beam beam surface surface concrete concrete strain strain perpe perpendic ndicular ular to diago diagonal nal concrete concrete struts angle angle of of concre concrete te diagonal diagonal strut strut in space space truss truss analo analogy gy ratio ratio of nominalnominal-to-c to-crack racking ing torques torques taken taken large largerr than than 1


REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-95) and Commentary (318R-95),” American Concrete Institute, Farmington Hills, Mich., 1995, 369 pp. 2. Hsu, T. T. C., “Torsion of Structural Concrete Behavior of Reinforced Concrete Rectangular Members,” Torsion of Structural Concrete, Concrete, SP-18, American Concrete Institute, Farmington Hills, Mich., 1968, pp. 261-306. 3. Lampert, P., and Thurlimann, B., “Torsion Tests of Reinforced Concrete Beams (Torsionsversuche an Stahlbetonbalken),” Report No. Report No. 6506-2, June 1968, 101 pp. 4. Ghoneim, M. G., and MacGregor, J. G., “Evaluation of Design Procedures for Torsion in Reinforced and Prestressed Concrete,” Report No. 184, Department of Civil Engineering, University of Alberta, Edmonton, Feb. 1993, 301 pp. 5. Rasmussen, L. J., and Baker, G., “Torsion in Reinforced Normal and High-Strength Concrete Beams—Part 1: Experimental Results,” ACI Structural Journal, Journal , V. 92, No. 1, Jan.-Feb. 1995, pp. 55-63. 6. Koutchoukali, N., “Non-Linear Behavior of Reinforced Concrete Beams Subjected to Pure Torsion,” PhD dissertation, Department of Civil Engineering, University of Missouri-Rolla, 1998, 127 pp. 7. Koutchoukali, N., and Belarbi, A., “Effect of Concrete Strength on the Behavior of Reinforced Concrete Beams Subjected to Pure Torsion,” Proceedings of High Strength Concrete First International Conference, Conference , ASCE, 1997, pp. 38-51.


8. Goodspeed, C. H.; Vanikar, S.; and Cook, R., “High-Performance Concrete Defined for Highway Structures,” Concrete International, International , V. 18, No. 2, Feb. 1996, pp. 62-67. 9. Alkhardaji, T., “Experimental Investigation of the Shear Flow Zone in Torsional Members,” Master’s thesis, University of Missouri-Rolla, 1998, 180 pp. 10. Collins, M. P., and Lampert, P., “Redistribution of Moments at Cracking—The Key to Simpler Design?” Analysis of Structural Systems for Torsion, Torsion, SP-35, American Concrete Institute, Farmington Hills, Mich., 1973, pp. 343-383. 11. Hsu, T. T. C., and Huang, C. S., “Torsional Limit Design of Spandrel Beams,” ACI J OURNAL, Proceedings V. 74, No. 2, Feb. 1977, pp. 71-79. 12. Ali, M. A., and White, R. N., “Toward a Rational Approach for Design of Minimum Torsion Reinforcement,” ACI Struct ural Journal , V. 96, No. 1, Jan.-Feb. 1999, pp. 40-45. 13. Leonhardt, F.; Walther, R.; and Schelling, A., “Torsionsversuche an Stahlbetonbalken,” Bulletin No. 239, Deutscher Ausschuss fur Stahlbeton, Berlin, 1974, 122 pp. 14. Lampert, P., and Collins, M. P., “Torsion, Bending, and Confusion— An Attempt to Establish the Facts,” ACI J OURNAL, Proceedings V. 69, No. 8, Aug. 1972, pp. 500-504.

469

Torsion

Recommend Documents